# Introduction to coding theory (CMU, Spr 2010)

## April 7, 2010

### Lecture 20 summary

Filed under: Lecture summary — Venkat Guruswami @ 6:46 pm

We touched upon some aspects relating to the combinatorics of list decoding Reed-Solomon codes. We discussed the geometric intuition underlying Reed-Solomon decoding, and presented Sudan’s algorithm for polynomial reconstruction/list decoding RS codes. We analyzed it by optimizing individual degrees leading to an algorithm working for agreement parameter $t > 2 \sqrt{kn}$ where $k$ is the degree and $n$ is the number of points. By optimizing the $(1,k)$-weighted degree, we improved this to $t \ge \sqrt{2kn}$.

We motivated the method of using multiple zeroes in the interpolation with an example, and then presented the details of how to enforce multiple (say $w$) zeroes at a point for a bivariate polynomial, why it buys us $w$ zeroes for each point of agreement, and why this should improve the agreement $t$ needed by a further $\sqrt{2}$ factor thus matching the Johnson bound.